Characterizations of Exponentiated Distributions

(1)

Gholam Hossein Hamedani

Marquette University United States of America

Abstract

Various characterizations of the class of exponentiated distributions are presented. These characterizations are based on a simple relationship between two truncated moments and based on functions of the nth order statistic. The results are applied to certain well-known members of this class.

1. Introduction

The problem of characterizing a distribution is an important problem which has recently attracted the attention of many researchers. Thus, various characterizations have been established in many different directions.

Recently, the Fα−distributions (or exponentiated distributions, defined below) have been shown to have a wide domain of applicability, in particular in modeling and analysis of life time data. Let F be an absolutely continuous cdf (cumulative distribution function) with support on

( )

a b, , where the interval may be unbounded, and let

α

be a

positive real number. The random variable X has an Fα −distribution if its cdf , denoted by FX, is given by FX

( )

x =F

( )

x = F x

( )

α

α _ _

  . The class of Fα−distributions contains certain well-known distributions for which their cdf 's have closed forms (see, e.g. Gupta et al. (1998), Gupta and Kundu (1999, 2000, 2001, 2007) and Nadarajah (2011)). For further details and the domain of applicability of this class, we refer the interested reader to the above mentioned articles and the most recent work of Shakil and Ahsanullah (2012) . In the statistical applications where the underlying distribution is assumed to be an Fα −distribution, the investigator needs to verify that this distribution is in fact the desired Fα −distribution. To this end the investigator has to rely on the characterizations of the underlying distribution and determine if the corresponding conditions are satisfied. So, the problem of characterizing Fα −distribution becomes essential. Our objective here, is to present characterizations of Fα −distributions. We shall do this in two different directions as discussed in Section 2 below.

The pdf (probability density function), f_X , of an Fα −distribution, F_X , with base cdf F , is given by

( )

( ) ( )

(

)

1

( )

= , < < , 1.1

X

f x αf x F x α− a x b

(2)

2. Characterization Results

As we mentioned in the previous section, the Fα −distributions have applications in many fields of study, which have been mentioned in Shakil and Ahsanullah (2012) and the references therein. So, an investigator will be vitally interested to know if their model fits the requirements of Fα−distribution. To this end, the investigator relies on characterizations of this distribution, which provide conditions under which the underlying distribution is indeed an Fα −distribution. In this section we will present various characterizations of the Fα−distributions.

Throughout this section we assume, when necessary, that the base distribution F is twice differentiable on its support.

2.1. Characterization based on two truncated moments

In this subsection we present characterizations of the Fα −distributions in terms of truncated moments. We like to mention here the works of Galambos and Kotz (1978), Kotz and Shanbhag (1980), Glänzel (1988, 1987 and 1990), Glänzel et al. (1994), Glänzel et al. (1984), Glänzel and Hamedani (2001) and Hamedani (1993, 2002 and 2006) in this direction. Our characterization results presented here will employ an interesting result due to Glänzel (1987) (Theorem G below).

Theorem G. Let

(

Ω, ,P

)

be a given probability space and let H =

[ ]

a b, be an interval for some a<b

(

a=−∞, =b ∞mightaswellbeallowed .

)

Let X:Ω →H be a continuous random variable with the distribution function F and let g and h be two real functions defined on H such that

( )

| =

( )

|

g X X ≥x h X X ≥x

   

   

E E η

( )

x , x∈H ,

is defined with some real function η . Assume that g , h∈C1

( )

H , η∈C2

( )

H and F is twice continuously differentiable and strictly monotone function on the set H . Finally, assume that the equation hη =g has no real solution in the interior of H . Then F is uniquely determined by the functions g , h and η , particularly

( )

=

( ) ( ) ( )

( )

exp

(

( )

)

,

' x

a

u

F x C s u du

u h u g u

η

η − −

∫

where the function s is a solution of the differential equation =

'

' h

s

h g

η

η − and C is a

constant, chosen to make = 1

HdF

∫

.

(3)

Proposition 2.1.2. Let X:Ω →

( )

a b, be a continuous random variable, F an absolutely continuous cdf with pdf f and let h x

( )

= 1 and

( )

= x

a

g x

∫

( ) ( )

(

)

1

f u F u α − du for x∈

( )

a b, . The pdf of X is

( )

1.1 if and only if the function η defined in Theorem G has the form

( )

1 1

( ) ( )

(

)

1

= ), < < .

2 2

x a

x f u F u α du a x b

η

α

−

+

∫

Proof. Let X have pdf

( )

1.1 , then

( )

(

1−F_X x

)

E_h X

( )

|X ≥x_= 1

(

−F_X

( )

x

)

, a<x< ,b and

( )

(

)

( )

( ) ( )

(

)

( )

( ) ( )

(

( )

)

1 2 1 | = 1 1

= = 1 , < < ,

2 X b t X x a b X X x

F x g X X x

f u F u duf t dt

F t f t dt F x a x b

α α α − − _ ≥ _ −

∫ ∫

∫

E and finally

( ) ( ) ( )

1 1

( ) ( )

(

)

1

= 0 < < .

2

x a

x h x g x f u F u α du for a x b

η α −   − _ − _≠ 

∫



Conversely, if η is given as above, then

( )

( ) ( ) ( )

( ) ( )

(

)

( ) ( )

(

)

1 1 = = , 1 ' ' x a

f x F x

x h x s x

x h x g x

f u F u du

α α η η α − − −  ₋    

∫

 and hence

( )

1

( ) ( )

(

)

1

= ln x , < < .

a

s x f u F u α du a x b

α

−

 

− _ − _



∫



Now, in view of Theorem G (with C=α) , X has pdf

( )

1.1 .

Corollary 2.1.3. Let X :Ω →

( )

a b, be a continuous random variable and let h x

( )

= 1 for x∈

( )

a b, . The pdf of X is

( )

1.1 if and only if there exist functions g and

η defined in Theorem G satisfying the differential equation

( )

( ) ( )

(

)

1

= , < < .

1

'

x a

f x F x

x

a x b

x g x

f u F u du

α α η η α − − −  ₋    

∫



Remark 2.1.4. The general solution of the differential equation given in Corollary 2.1.3 is

( )

1

( ) ( )

(

)

1 1

( ) ( ) ( )

(

)

1

= x ,

x f u F u α du g x f x F x α dx D

η

−

− −

 ₋ _{ }₋ ₊ _

(4)

for a<x<b , where D is a constant. One set of appropriate functions is given in

Proposition 2.1.2 with = 1₂. 2 D

α

Remark 2.1.5. Due to the generality of the expression for the Fα −distributions (being a power of a given base cdf ), the statement of Proposition 2.1.2 is given in a general

form in which we assume h x

( )

= 1 for x∈

( )

a b, . We will, however, take different

( )

h x and g x

( )

for the special cases that we characterize below in the interest of having

( )

x

η as simple as possible.

( )

I Generalized Exponential (GE) or Exponentiated Exponential (EE) Distribution.

The cdf and pdf for this subclass of Fα −distributions are given , respectively, by

( )

(

)

( )

(

)

1

= = 1 , 0,

= 1 , > 0,

x X

x x

X

F x F x e x

f x e e x

α

α _λ

α

λ λ

αλ

−

− −

− ≥

 

 

−

where λ , α are positive parameters.

Proposition 2.1.2 with h x

( )

= 1

(

−e−λx

)

− −(α 1) , g x

( )

= x

(

1−e−λx

)

− −(α 1) and

( )

1

=

x x

η

λ

+ , gives a characterization of this subclass of distributions.

( )

II Exponentiated Weibull (EW) Distribution.

The cdf and pdf for this subclass of Fα−distributions are given , respectively, by

( )

(

)

( )

(

)

1

= = 1 , 0,

= 1 , > 0,

x X

x x

X

F x F x e x

f x x e e x

α δ

α _λ

α

δ δ

δ λ λ

αλδ

−

− − −

− ≥

 

 

−

where λ ,

α

, δ are positive parameters.

Proposition 2.1.2 with

( )

(

)

( 1)

= 1 x

h x e

α δ

λ − −

−

− ,

( )

(

)

( 1)

= 1 x

g x x e

α δ

λ − −

−

− and

( )

= 1

x

e

x x

δ λ

δ

η

δλ

+ λxδ;1

δ

 

Γ_ _ where

( )

; = 1 v

u

u β ∞vβ− −e dv

Γ

∫

, gives a characterization of

this subclass of distributions.

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( )

III Exponentiated Pareto or Lomax (EP or EL) Distribution.

The cdf and pdf for this subclass of Fα−distributions are given , respectively, by

( )

(

)

( )

₁

(

)

( 1)

= = , 0,

= , > 0,

X

F x F x x x x

f x x x x

α

α _αδ _δ

α

αδ δ

λ

αδλ λ

−

− + −

+ ≥

 

 

+

where λ ,

α

, δ are positive parameters.

Let γ be an arbitrary positive real number. Proposition 2.1.2 with δ γ> ,

( )

₍ ₁₎

(

)

( 1)

=

h x x− +α δ xδ +λ α+ , g x

( )

=x− +(α 1)δ γ+

(

xδ +λ

)

(α−1), and η

( )

x = δ xγ

δ γ

 

 ₋ 

  ,

gives a characterization of this subclass of distributions. Another version of (EP or EL)

has cdf FX

( )

x = 1

(

1 x

)

,

α β −

 _{− +} 

  x≥0 for which h x

( )

, g x

( )

and η

( )

x can be

defined as well.

( )

IV Generalized Generalized Logistic (GGL) Distribution.

The cdf and pdf for this subclass of Fα −distributions are given , respectively, by

( )

(

)

( )

(

)

( 1)

1

= 1 , ,

x X

x x

X

F x e x

f x x e e x

α δ λ

α

δ δ

δ λ λ

αδλ

− −

− +

− − −

+ ∈

R

where λ ,

α

> 0 and δ a positive odd integer, are parameters.

Proposition 2.1.2 with

( )

(

)

( 1)

= 1 x

h x e

α δ

λ +

−

+ ,

( )

(

)

( 1)

= 1 x

g x x e

α δ

λ +

−

+ , and

( )

= 1

x

e

x x

δ λ

δ

η

δλ

+ λxδ;1

δ

 

Γ_ _, gives a characterization of this subclass of distributions.

For δ = 1 , the corresponding distribution is called Generalized Logistic (GL), so for δ a positive odd integer we call the corresponding distribution (GGL).

Remark 2.1.6. Clearly there are other suitable functions h x

( )

, g x

( )

and η

( )

x for the above special cases

( ) ( )

I − IV .

2.2. Characterization based on truncated moments of certain functions of the nth order statistic

Let X1:n ≤X2:n ≤ ≤... Xn n: be n order statistics from an F

α ₋

distribution with base

cdf F. We present here a characterization result for Fα −distributions based on truncated moments of the th

(6)

characterizations of other well-known continuous distributions in this direction. The proof of the following proposition is similar to that of Theorem 2.5 of Hamedani (2010) in which k x

( )

=xδ for some δ > 0. We give a brief proof of the general k x

( )

, however, for the sake of completeness.

Proposition 2.2.1. Let X:Ω →

( )

a b, be a continuous random variable with Fα − distribution F_X and k x

( )

be a differentiable function such that

( )

(

( )

)

= 0. lim

n

x→ak x FX x Let q x n

( )

, be a real-valued function which is differentiable

with respect to x and

( )

, =

' b a

k x dx

q x n ∞

∫

. Then

[

k X( n n: ) |Xn n: <t

]

=k t

( ) ( )

−q t n, , a< < ,t b

(

2.2.1

)

E

implies that

( )

1

,

= exp , .

, ,

'

n _b

X _x

q b n k t

F x dt x a

q x n nq t n

  _ _

− ≥

   

  _ _

  

∫



Proof. Condition

(

2.2.1 and the assumption

)

limx→ak x

( )

(

FX

( )

x

)

n = 0 imply

t '

( )

(

_X

( )

)

n =

( )

,

(

_X

( )

)

n.

(

2.2.2

)

ak x F x dx q t n F t

∫

Differentiating

(

2.2.2 with respect to

)

t , we obtain

( )

(

)

,

= . 2.2.3

, ,

' X

X

q t n

f t _t k t

F t nq t n nq t n

∂ ∂ +

Integrating

(

2.2.3 with respect to

)

t , from x to b , results in

( )

1

,

= exp , .

, ,

'

n _b

X _x

q b n k t

F x dt x a

q x n nq t n

  _ _

− ≥

   

  _ _

  

∫



Remarks 2.2.2.

( )

i In Proposition 2.2.1 , the interval

( )

a b, is allowed to be unbounded, as we mentioned in the Introduction.

( )

ii In Hamedani (2010), the author did not have any applications of their Theorem 2.5. We are now pleased to see that there are distributions for which Proposition 2.2.1 (or Theorem 2.5) can be employed to characterize them. We now present characterizations of Fα −distributions

( ) ( )

I − IV

(see our Remark 2.1.5) based on certain functions of the th

n order statistic, X_{n n}_: . It is rather straightforward to show that the conditions of Proposition 2.2.1 are satisfied in cases

( )

I ,

( )

II and

( )

IV . The case

( )

III requires rewriting FX

( )

x in slightly

different form ( F_X

( )

x = 1

(

x

)

1

)

α δ

λ λ −

(7)

( )

I For k x

( )

= 1

(

−e−λx

)

α and

( )

, = 1

(

1

)

1

x

q x n e

n

α λ − −

+ ,

(

2.2.1 provides a

)

characterization of (GE or EE) distributions. For

( )

= 1

(

x

)

k x −e−λ and

( )

1

(

)

, = 1

1

x

q x n e

n

λ

α+ − − ,

(

2.2.1

)

provides a characterization of (GE or EE)

distributions as well.

( )

II For k x

( )

= 1

(

e x

)

α δ λ −

− and

( )

, = 1

(

1

)

1

x

q x n e

n

α δ λ − −

+ ,

(

2.2.1 provides a

)

characterization of (EW) distributions. As expected the functions k x

( )

= 1

(

−e−λxδ

)

and

( )

, = 1

(

1

)

1

x

q x n e

n

λ

α+ − − will work as well.

( )

III For k x

( )

=

(

xδ +λ

)

−1 and

( )

(

)

(

)

1

, = 1

1

q x n x x

n

δ δ

λ α

−

− +

+ ,

(

2.2.1 provides

)

a characterization of (EP or EL) distributions.

( )

IV For

( )

(

)

1

= 1 x

k x e λδ

− −

+ and

( )

(

)

1

, = 1

1

x

q x n e

n

δ λ

α

− − +

+ ,

(

2.2.1 provides a

)

characterization of (GGL) distributions.

References

1. Ahsanullah, M. and Hamedani, G.G.

(

2007 , Certain characterizations of the

)

power function and beta distributions based on order statistics, JSTA, 6, 220 225.−

2. Galambos, J. and Kotz, S.

(

1978 , Characterizations of probability distributions.

)

A unified approach with an emphasis on exponential and related models, Lecture Notes in Mathematics, 675, Springer, Berlin.

3. Glänzel, W.

(

1988 , A characterization of the normal distribution, Studia Sci.

)

Math. Hungar. 23, 89 91.−

4. Glänzel, W.

(

1987 , A characterization theorem based on truncated moments and

)

its application to some distribution families, Mathematical Statistics and Probability Theory (Bad Tatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht,

75 84.−

5. Glänzel, W.

(

1990 , Some consequences of a characterization theorem based on

)

truncated moments, Statistics 21, 613 618.−

(8)

7. Glänzel, W., Telcs, A. and Schubert, A.

(

1984 . Characterization by truncated

)

moments and its application to Pearson-type distributions, Z. Wahrsch. Verw. Gebiete 66, 173 183.−

8. Glänzel, W. and Hamedani, G.G.

(

2001 . Characterizations of univariate

)

continuous distributions, Studia Sci. Math. Hungar. 37 , 83 118.−

9. Gupta, R.C., Gupta, P.L. and Gupta, R.D.

(

1998 . Modeling failure time data by

)

Lehman alternatives, Commun. Statist.-Theor. Meth., 27 , 887 904.−

10. Gupta, R.D. and Kundu, D.

(

1999 . Generalized exponential distribution,

)

Austral. & New Zealand J. Statist., 41, 173 188.−

(

2000 . Generalized exponential distribution:

)

different method of estimations, J. Statist. Comput. Simul., 00, 1 22.−

(

2001 . Exponentiated exponential family: an

)

alternative to gamma and Weibull distributions, Biometrical J., 43, 117 130.−

(

2007 . Generalized exponential distribution:

)

Existing results and some recent developments, J. Statist. Plann. Inference, 137,

3537 3547.−

14. Hamedani, G.G.

(

1993

)

. Characterizations of Cauchy, normal and uniform distributions, Studia Sci. Math. Hungar. 28, 243 247.−

15. Hamedani, G.G.

(

2002

)

. Characterizations of univariate continuous distributions.

II, Studia Sci. Math. Hungar. 39, 407 424.−

16. Hamedani, G.G.

(

2006

)

. Characterizations of univariate continuous distributions.

III, Studia Sci. Math. Hungar. 43, 361 385.−

17. Hamedani, G.G.

(

2010 . Characterizations of continuous univariate distributions

)

based on the truncated moments of functions of order statistics, Studia Sci. Math. Hungar. 47, 462 484.−

18. Hamedani, G.G. and Ahsanullah

(

2005 . Characterizations of univariate

)

continuous distributions based on hazard function II, JSTA, 4, 218 238.−

19. Hamedani, G.G., Ahsanullah, M. and Sheng, R.

(

2008 . Characterizations of

)

certain continuous univariate distributions based on truncated moment of the first order statistic, Aligrah J. of Statistics, 28, 75 81.−

20. Kotz, S. and Shanbhag, D.N.

(

1980 . Some new approaches to probability

)

distributions, Adv. in Appl. Probab. 12, 903 921.−

21. Nadarajah, S.

(

2011 . The exponentiated exponential distribution: a survey,

)

AStA Advances in Statistical Analysis, 95 , 219 251.−

22. Shakil, M. and Ahsanullah, M.

(

2012 . Review on order statistics and record

)